Frattini subgroup: Difference between revisions

Latest revision as of 06:49, 20 April 2017

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

QUICK PHRASES: intersection of all maximal subgroups, biggest subgroup contained in every maximal subgroup

Let $G$ be a group. The Frattini subgroup of $G$ , denoted $\Phi (G)$ , is defined in the following equivalent ways:

It is the intersection of all subgroups $M\leq G$ , where $M$ is maximal in $G$
It is the set of all nongenerators, i.e., elements $x\in G$ such that if $S\cup \{x\}$ is a generating set for $G$ , then so is $S$

When $G$ is a group in which every subgroup is contained in a maximal subgroup, then the Frattini subgroup is also the unique largest Frattini-embedded normal subgroup.

Equivalence of definitions

Further information: Equivalence of definitions of Frattini subgroup

Relation with other subgroup-defining functions

Larger subgroup-defining functions

Fitting subgroup always contains the Frattini subgroup (for a finite group) This follows from the fact that the Frattini subgroup is nilpotent.

Smaller subgroup-defining functions

Derived subgroup when the whole group is nilpotent. For full proof, refer: Nilpotent implies derived in Frattini

Effect of operators

Fixed-point operator

A group equals its own Frattini subgroup if and only if it has no maximal subgroups. For instance, the group of rational numbers, or more generally, the additive group of any field of characteristic zero has no maximal subgroups.

Free operator

A group whose Frattini subgroup is trivial is termed a Frattini-free group.

Group properties satisfied

In general, it is hard to find group properties satisfied by the Frattini subgroup of every group. However, for a group in which every subgroup is contained in a maximal subgroup, the Frattini subgroup usually satisfies some fairly strong restrictions. Most of these are restrictions that are satisfied by any Frattini-embedded normal subgroup. These include:

Inner-in-automorphism-Frattini group: For full proof, refer: Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini
ACIC-group: For full proof, refer: Frattini-embedded normal-realizable implies ACIC

Since any finite ACIC-group is nilpotent, the Frattini subgroup of any finite group is nilpotent. For full proof, refer: Frattini subgroup is nilpotent

Also related are:

Subgroup properties satisfied

The Frattini subgroup of any group is a characteristic subgroup.

For a finite group, the Frattini subgroup is always a proper subgroup (because there exist maximal subgroups).

Subgroup-defining function properties

Monotonicity

The Frattini subgroup-defining function is not monotone. However, the Frattini subgroup of any normal subgroup is contained in the Frattini subgroup of the whole group, provided that normal subgroup satisfies the property that every proper subgroup is contained in a maximal subgroup. In particular, for any finite group, the Frattini subgroup of a normal subgroup is contained in the Frattini subgroup of the whole group.

For full proof, refer: Frattini subgroup is normal-monotone

Idempotence and iteration

The Frattini subgroup-defining function is not idempotent.

Quotient-idempotence

This subgroup-defining function is quotient-idempotent: taking the quotient of any group by the subgroup, gives a group where the subgroup-defining function yields the trivial subgroup
View a complete list of such subgroup-defining functions

Associated constructions

Associated quotient-defining function

The quotient-defining function associated with this subgroup-defining function is: Frattini quotient

Associated descending series

The associated descending series to this subgroup-defining function is: Frattini series

The Frattini series is the series obtained by iterating the Frattini subgroup operation, starting with the whole group. It gives aa descending series.

If this series terminates at the identity in finite length (which it will for a finite group, since the Frattini subgroup at each stage will be proper) then the length of the series is termed the Frattini length for the group.

Computation

GAP command

The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:FrattiniSubgroup
View other GAP-computable subgroup-defining functions

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, ^{More info}, Page 198-199
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613, ^{More info}, Page 135
Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 173, Section 5.1 (definition in paragraph, preceding Theorem 1.1)

@@ Line 2: / Line 2: @@
 ==Definition==
+{{quick phrase|intersection of all maximal subgroups, biggest subgroup contained in every maximal subgroup}}
-===Symbol-free definition===
+Let <math>G</math> be a [[group]]. The '''Frattini subgroup''' of <math>G</math>, denoted <math>\Phi(G)</math>, is defined in the following equivalent ways:
-The '''Frattini subgroup''' of a group is defined in the following equivalent ways:
+# It is the [[defining ingredient::intersection of subgroups|intersection]] of all subgroups <math>M \le G</math>, where <math>M</math> is [[maximal subgroup|maximal]] in <math>G</math>
+# It is the set of all [[defining ingredient::nongenerator]]s, i.e., elements <math>x \in G</math> such that if <math>S \cup \{ x \}</math> is a generating set for <math>G</math>, then so is <math>S</math>
-* The intersection of all its [[maximal subgroup]]s
+When <math>G</math> is a [[group in which every subgroup is contained in a maximal subgroup]], then the Frattini subgroup is also the unique largest [[defining ingredient::Frattini-embedded normal subgroup]].
-* The set of all [[nonegenerator]]s
-==Properties of the image==
+===Equivalence of definitions===
-===Group properties satisfied===
+{{further|[[Equivalence of definitions of Frattini subgroup]]}}
-The Frattini subgroup of any finite group is a [[nilpotent group]]. This follows from [[Frattini's argument]]. {{proofat|[[Frattini subgroup is nilpotent]]}}
+==Relation with other subgroup-defining functions==
-===Subgroup properties satisfied===
+===Larger subgroup-defining functions===
+* [[Fitting subgroup]] always contains the Frattini subgroup (for a [[finite group]]) This follows from the fact that the [[Frattini subgroup is nilpotent]].
+===Smaller subgroup-defining functions===
+* [[Derived subgroup]] when the whole group is nilpotent. {{proofat|[[Nilpotent implies derived in Frattini]]}}
+==Effect of operators==
+===Fixed-point operator===
+A group equals its own Frattini subgroup if and only if it has [[group without maximal subgroups|no maximal subgroups]]. For instance, the [[group of rational numbers]], or more generally, the additive group of any field of characteristic zero has no maximal subgroups.
+===Free operator===
+A group whose Frattini subgroup is trivial is termed a [[Frattini-free group]].
+==Group properties satisfied==
+In general, it is hard to find group properties satisfied by the Frattini subgroup of every [[group]]. However, for a group in which every subgroup is contained in a maximal subgroup, the Frattini subgroup usually satisfies some fairly strong restrictions. Most of these are restrictions that are satisfied by ''any'' [[Frattini-embedded normal subgroup]]. These include:
+* [[Inner-in-automorphism-Frattini group]]: {{proofat|[[Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini]]}}
+* [[ACIC-group]]: {{proofat|[[Frattini-embedded normal-realizable implies ACIC]]}}
+Since any finite ACIC-group is [[nilpotent group|nilpotent]], the Frattini subgroup of any finite group is nilpotent. {{proofat|[[Frattini subgroup is nilpotent]]}}
+Also related are:
+* [[:Category:Terminology related to realization problems for Frattini subgroup]]
+* [[:Category:Facts related to realization problems for Frattini subgroup]]
+==Subgroup properties satisfied==
 The Frattini subgroup of any group is a [[characteristic subgroup]].
@@ Line 26: / Line 59: @@
 ===Monotonicity===
-{{normal-monotone sdf}}
+The Frattini [[subgroup-defining function]] is not monotone. However, the Frattini subgroup of any [[normal subgroup]] is contained in the Frattini subgroup of the whole group, provided that normal subgroup satisfies the property that [[group in which every proper subgroup is contained in a maximal subgroup|every proper subgroup is contained in a maximal subgroup]]. In particular, for any [[finite group]], the Frattini subgroup of a normal subgroup is contained in the Frattini subgroup of the whole group.
-The Frattini [[subgroup-defining function]] is not monotone. However, the Frattini subgroup of any [[normal subgroup]] is contained in the Frattini subgroup of the whole group. Hence, it is a [[normal-monotone subgroup-defining function]].
 {{proofat|[[Frattini subgroup is normal-monotone]]}}
@@ Line 34: / Line 65: @@
 ===Idempotence and iteration===
-The Frattini subgroup-defining function is not idempotent. Iteration of the Frattini subgroup-defining function gives the [[Frattini series]] of a group. For any finite nontrivial group, the Frattini subgroup is always [[proper subgroup|proper]] and hence the Frattini series terminates after a finite length at the trivial subgroup.
+The Frattini subgroup-defining function is not idempotent.
+{{quotient-idempotent sdf}}
+==Associated constructions==
+{{associated qdf|Frattini quotient}}
+{{associated ds|Frattini series}}
+The Frattini series is the series obtained by iterating the Frattini subgroup operation, starting with the whole group. It gives aa descending series.
+If this series terminates at the identity in finite length (which it will for a finite group, since the Frattini subgroup at each stage will be proper) then the length of the series is termed the [[Frattini length]] for the group.
-===Quotient-idempotence and quotient-iteration===
+==Computation==
-The quotient function associated with the Frattini subgroup is idempotent.
+{{GAP command for sdf|FrattiniSubgroup}}
+==References==
+===Textbook references===
+* {{booklink-defined|DummitFoote}}, Page 198-199
+* {{booklink-defined|RobinsonGT}}, Page 135
+* {{booklink-defined|Gorenstein}}, Page 173, Section 5.1 (definition in paragraph, preceding Theorem 1.1)